Theorem xrmaxleim 11988

Description: Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)

Assertion

Ref	Expression
xrmaxleim	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Proof of Theorem xrmaxleim

Dummy variables f g y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrlttri3 10178	. . . 4 |- ( ( f e. RR* /\ g e. RR* ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 277	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3		simplr 533	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. RR* )
4		prid2g 3812	. . . 4 |- ( B e. RR* -> B e. { A , B } )
5	3, 4	syl 14	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. { A , B } )
6		simpllr 540	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> A <_ B )
7		xrlenlt 8380	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
8	7	ad3antrrr 496	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( A <_ B <-> -. B < A ) )
9	6, 8	mpbid 147	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < A )
10		breq2 4129	. . . . . . 7 |- ( y = A -> ( B < y <-> B < A ) )
11	10	notbid 677	. . . . . 6 |- ( y = A -> ( -. B < y <-> -. B < A ) )
12	11	adantl 277	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( -. B < y <-> -. B < A ) )
13	9, 12	mpbird 167	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < y )
14		xrltnr 10160	. . . . . 6 |- ( B e. RR* -> -. B < B )
15	14	ad4antlr 499	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < B )
16		breq2 4129	. . . . . . 7 |- ( y = B -> ( B < y <-> B < B ) )
17	16	notbid 677	. . . . . 6 |- ( y = B -> ( -. B < y <-> -. B < B ) )
18	17	adantl 277	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> ( -. B < y <-> -. B < B ) )
19	15, 18	mpbird 167	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < y )
20		elpri 3728	. . . . 5 |- ( y e. { A , B } -> ( y = A \/ y = B ) )
21	20	adantl 277	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> ( y = A \/ y = B ) )
22	13, 19, 21	mpjaodan 810	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> -. B < y )
23	2, 3, 5, 22	supmaxti 7334	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> sup ( { A , B } , RR* , < ) = B )
24	23	ex 115	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   {cpr 3706   class class class wbr 4125   supcsup 7312   RR*cxr 8349   < clt 8350   <_ cle 8351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltirr 8281  ax-pre-apti 8284

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-iota 5332  df-riota 6028  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356

This theorem is referenced by:  xrmaxltsup  12002  xrmaxadd  12005  xrmineqinf  12013